Properties

Label 4-245e2-1.1-c3e2-0-10
Degree $4$
Conductor $60025$
Sign $1$
Analytic cond. $208.960$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 10·3-s − 2·4-s − 10·5-s − 20·6-s + 21·9-s − 20·10-s + 66·11-s + 20·12-s − 10·13-s + 100·15-s − 20·16-s − 70·17-s + 42·18-s − 140·19-s + 20·20-s + 132·22-s − 16·23-s + 75·25-s − 20·26-s + 310·27-s − 258·29-s + 200·30-s + 20·31-s − 200·32-s − 660·33-s − 140·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.92·3-s − 1/4·4-s − 0.894·5-s − 1.36·6-s + 7/9·9-s − 0.632·10-s + 1.80·11-s + 0.481·12-s − 0.213·13-s + 1.72·15-s − 0.312·16-s − 0.998·17-s + 0.549·18-s − 1.69·19-s + 0.223·20-s + 1.27·22-s − 0.145·23-s + 3/5·25-s − 0.150·26-s + 2.20·27-s − 1.65·29-s + 1.21·30-s + 0.115·31-s − 1.10·32-s − 3.48·33-s − 0.706·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60025\)    =    \(5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(208.960\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{245} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 60025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
3$C_2$ \( ( 1 + 5 T + p^{3} T^{2} )^{2} \)
11$D_{4}$ \( 1 - 6 p T + 325 p T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 70 T + 6651 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 140 T + 14218 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 16 T + 15774 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 20 T + 20082 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 328 T + 106906 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 300 T + 50342 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 30 T + 190271 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 540 T + 212254 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 380 T + 429258 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1080 T + 705962 T^{2} + 1080 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 468 T + 554906 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 860 T + 522934 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 40 T + 703974 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 240 T - 164062 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55639106780278446091833485974, −11.17640015452778181744863073085, −10.73759722190597981294620729364, −10.34679310526481175010457988521, −9.400656195664871413315169362982, −8.882492241037928788796798324180, −8.688520042377446474929939395402, −7.84200990705642548892143387115, −7.11204992735264282915461643907, −6.60332938451781971481346140290, −6.22942758318084413538326499999, −5.86149798837209534943728765147, −4.86943917233339096392053040151, −4.81715488063016183995797723016, −4.00443109216199791205280469390, −3.78191278706345279968320018666, −2.47739258038033237411274477461, −1.31984406205137736308868858378, 0, 0, 1.31984406205137736308868858378, 2.47739258038033237411274477461, 3.78191278706345279968320018666, 4.00443109216199791205280469390, 4.81715488063016183995797723016, 4.86943917233339096392053040151, 5.86149798837209534943728765147, 6.22942758318084413538326499999, 6.60332938451781971481346140290, 7.11204992735264282915461643907, 7.84200990705642548892143387115, 8.688520042377446474929939395402, 8.882492241037928788796798324180, 9.400656195664871413315169362982, 10.34679310526481175010457988521, 10.73759722190597981294620729364, 11.17640015452778181744863073085, 11.55639106780278446091833485974

Graph of the $Z$-function along the critical line